Equilibrium fluctuation of the Atlas model

We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite (Z+-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on R+. In this context, we show that the joint law of ranked particles, after being centered and scaled by t.14, converges as t.. to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on R+ with the Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional Brownian Motion (fBM). In particular, we prove a conjecture of Pal and Pitman [Ann. Appl. Probab. 18 (2008) 2179.2207] about the asymptotic Gaussian fluctuation of the ranked particles.